Kim said there had been repeated crossings of the demarcation line by North Korean soldiers “while installing tactical roads, fences, and laying mines”.
South Korean troops fired warning shots and issued broadcasts to encourage the North Koreans to withdraw to their side, he said.
Kim said the recent incursions came about because of “the loss of many MDL markers” that were installed under the 1953 Armistice Agreement that brought a halt to the Korean War.
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Seoul and Pyongyang technically remain at war because the conflict ended in an armistice and not a peace treaty.
The MDL lies inside the Demilitarised Zone (DMZ), a four-kilometre-wide buffer zone that runs for 250 kilometres (160 miles) across the Korean peninsula.
The proposed military talks follow South Korean President Lee Jae Myung’s offer of broader discussions with the North without preconditions, a sharp reversal from the hawkish stance taken by his conservative predecessor.
Lee has taken several steps to ease military tensions with the nuclear-armed North since his inauguration in June, including removing propaganda loudspeakers along the border and banning the dropping of anti-Pyongyang leaflets.
Pyongyang has yet to respond to Lee’s overtures.
Former president Yoon Suk Yeol adopted a hardline approach to the North, which has drawn increasingly close to Moscow following Russia’s invasion of Ukraine, that pushed inter-Korean relations to one of their lowest points in years.
The two Koreas engaged in a tit-for-tat propaganda war last year, with the North sending thousands of trash-filled balloons southwards in retaliation for propaganda balloons launched by South Korean activists.